To solve the equations you've provided, you can use several mathematical properties, including the distributive property, the properties of equality, and the addition and multiplication properties. Here’s a summary of the properties you might use for each equation:
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Distributive Property: Use this to simplify the left side: \[ 2(3 - 7x) + 4x = 8 - 5x \] will become \[ 6 - 14x + 4x = 8 - 5x. \]
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Combining Like Terms: In this equation, you can combine like terms on both sides: \[ 6 - 14x + 4x = 8 - 5x \] simplifies to \[ 6 - 10x = 8 - 5x. \]
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Moving Terms: You can use the properties of equality to move terms to one side: \[ 6 - 10x = 8 - 5x \] can be rearranged to isolate \( x \).
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Subtracting from Both Sides: You can subtract 6 from both sides: \[ 6 - 5x = 8 \] becomes \[ -5x = 8 - 6. \]
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Dividing by a Constant: To isolate \( x \): \[ 5x = 2 \] can be solved by dividing both sides by 5.
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Understanding Negative Values: This step shows a simple calculation of division: \[ x = -\frac{2}{5}. \]
In summary, the properties you would use include:
- Distributive Property for expanding expressions.
- Combining Like Terms for simplifying expressions.
- Properties of Equality (addition and subtraction) for manipulating equations to isolate variables.
- Division Property of Equality for solving for the variable.
By applying these properties step-by-step, you can solve each equation systematically.